2016 CoSMo - CompNeuroscicompneurosci.com/wiki/images/8/83/Francisco_2016_CoSMo.pdf · Frame computational neuromechanics as a Big Data problem. Hot topics of the day Muscle redundancy

CoSMo 2016

Francisco Valero-Cuevas, PhD Brain-Body Dynamics Laboratory

Purpose of this talk

To be informative, but also provocative, about the role of mechanics and

computational science understanding the control neuromuscular systems.

Provide several examples of how mechanics has helped us understand neuroscience.

Frame computational neuromechanics as a Big Data problem.

Hot topics of the day

Muscle redundancyMuscle Synergies

Optimization and Optimal controlMotor learningMotor policies

Bayesian motor control

My scientific approach to these problems

Neurophysiology

Sensorimotor behavior

Musculoskeletal modeling

Theories of neural control

Nylon cordsdriving tendons

Cadaverichand

Physical modeling

Clinical and rehabilitation tools

Machine learning

x

x

τ!"#$%&$

'()$*

!

+

,*-".

/$$)0-12

3-4"

Some long term questions How does the neuromechanical system (or how should a versatile machine) meet the necessary

conditions for dexterous function?

and/or

What specific contributions from passive (e.g., the body) and active (e.g., the brain) components

enable dexterous function?

For example, computational modeling of neuromuscular systems is still in its infancy

Robust Control

Adaptive control

Hierarchical control

Hybrid control

Model predictive control

Reinforcementlearning

(Policy gradients,Q-learning)

*Optimal Control(LQR, LQG, iLQR, iLQG)

*Stochastic Estimation(Kalman filter, extended Kalman filter)

*Supervised learning(Backpropagation ANNs,support vector machines)

*Unsupervised learning(PCA, self-organized ANNs)

*Bayesian estimation(Particle filters, path integral estimation,

Monte Carlo simulations)

**Classical control(Root locus design,

tracking)

Minimum variance unbiased estimation (MVUE)Best linear unbiased estimator (BLUE)

etc.

Machine learning Control theory

Estimation-Detection theoryValero-Cuevas et al., IEEE RBME 2009

The problems in motor neuroscience todayWe agree on physics, mechanics, physiology, and computational principles; but the consequences of our experimental and modeling choices are hard to reconcile.

As a result, we are united by our methods but fragmented into schools of thought that tend to talk and publish past each other.

Consider the longstanding debates around topics like internal models, equilibrium point control, optimal control, synergies, etc.

Many of these debates become a matter of opinion.

The fundamental challenge for organisms:Newton and Darwin are unforgiving

Mechanics describes theundeniable physical reality.

Evolutionary biology is the response to that reality. Organisms are a result of successful

brain-body co-evolution in that context.

We must move fromComputational Neuroscience

toComputational Neuromechanics

Where is one to begin?

From The Help by Kathryn Stockett, a novel about life in the ‘60s:…get an entry-level job... When you’re not making mimeographs ... look around, Don’t waste your time on the obvious things. Write about what disturbs you, particularly if it bothers no one else.

Some mechanics-based examples of things that have been disturbing some of us

Why do we have so many muscles?

Neuromechanical Concept #0

Control of force(underdetermined—redundancy)

vs.Control of motion

(overdetermined—no redundancy)

Simplest tendon mechanics

42 4 Tendon-Driven Limbs

r =sq

r

s

q q

Fig. 4.5 Schematic representation of the calculations of the moment arm, r(q). Given tendonexcursion as a function of joint angle s(q), the moment arm is its partial derivative with respect tothe joint angle r(q) = ds

dq , Eq. 4.9 [4]

Fig. 4.6 Measurement oftendon excursion for asimple tendon path. Notethat a negative (as per theright hand rule) rotation ofthe joint −δq induces apositive (rightward) tendonexcursion δs that lengthensthe musculotendon, and viceversa. See Sect. 4.6 for adefinition of this signcovention

linear scale

angular scale+ s

- q

Fig. 4.7 A planar 1 DOFlimb driven by 6 tendons

r3

r2

r1

m1

m3

m2

m4

m6

m5

Force control vs. movement control44 4 Tendon-Driven Limbs

τ = r(q)T fm (4.14)

and

τ =

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠·

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠=

(r(q)1, r(q)2, . . . , r(q)N

)T

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠(4.15)

This represents amapping from the vector spaceofmuscle forces (of dimensionN , or fm ∈ RN ) to the scalar value of net joint torque (of dimension 1, orτ ∈ R1),

RN → R1 (4.16)

Equations4.12–4.15 are underdetermined in that they equivalently express thata given value of the net joint torque scalar can be produced by a variety of combi-nations of muscle forces. Essentially, there are N muscle activation DOFs that canbe combined in an infinite number of ways to achieve a same goal.2 This is a formof muscle redundancy that begs the question of how the nervous system selects asolution from amongmany. This has been called the central problem ofmotor control[7]. We will discuss this in detail in Chap.5.

An equally important concept described by Fig. 4.7—which is not as well high-lighted in the literature [8]—concerns the tendon excursions of the N muscles thatcross the joint. In this case of a single DOF q

δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛

⎜⎜⎜⎝

δs1δs2...

δsN

⎞

⎟⎟⎟⎠=

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠δq (4.18)

which in vector form is

δs = −r(q) δq (4.19)

2Note that the letter M need not stand for muscles, nor N be used only for kinematic DOFs of alimb. They are simply letters to indicate indices and dimensions. See Appendix A.


τ = r(q)T fm (4.14)

and

τ =

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠·

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠=

(r(q)1, r(q)2, . . . , r(q)N

)T

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠(4.15)


RN → R1 (4.16)



δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛

⎜⎜⎜⎝

δs1δs2...

δsN

⎞

⎟⎟⎟⎠=

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠δq (4.18)


δs = −r(q) δq (4.19)


r3r2r1

m1

m3

m2

m4

m6

m5


τ = r(q)T fm (4.14)

and

τ =

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠·

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠=

(r(q)1, r(q)2, . . . , r(q)N

)T

⎛

⎜⎜⎜⎝

f1f2...

fN

⎞

⎟⎟⎟⎠(4.15)


RN → R1 (4.16)



δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛

⎜⎜⎜⎝

δs1δs2...

δsN

⎞

⎟⎟⎟⎠=

⎛

⎜⎜⎜⎝

r(q)1r(q)2...

r(q)N

⎞

⎟⎟⎟⎠δq (4.18)


δs = −r(q) δq (4.19)


underdetermined

overdetermined

Reflexes respond to tendon excursions

Motor Cotex

Motorneuron

Extensor

Flexor

MuscleSpindle

Sensory Cortex

Sensory Neuron

To lengthen a muscle….you have to silence its stretch reflex!


Feasible functionvs.

Optimal function

for underdetermined systems

Muscle redundancy as the central problem of motor control.

Popular View:We have many more muscles than kinematic degrees of freedom.

This allows infinite solutions.

Therefore the nervous system is faced with the tough computational problem of decision-making (or optimization).

But this is paradoxical withevolutionary biology and clinical reality

Wikipedia

That is,For decades, neuroscientists and biomechanists have

been exploring how to effectively choose specific muscle actions from a set of infinite choices.

but...

If we are so redundant:Which muscle would you like to donate?

Why do people seek clinical treatment for dysfunction even after mild pathology?

Why would we evolve, encode, grow, maintain, repair, control, etc. so many muscles?

1

ISBN 978-1-4471-6746-4

Biosystems & Biorobotics

Francisco J. Valero-Cuevas

Fundamentals of NeuromechanicsFundamentals of Neuromechanics

Valero-Cuevas


Francisco J. Valero-CuevasFundamentals of Neuromechanics

This book provides a conceptual and computational framework to study how the nervous system exploits the anatomical properties of limbs to produce mechanical function. The study of the neural control of limbs has historically emphasized the use of optimization to find solutions to the muscle redundancy problem. That is, how does the nervous system select a specific muscle coordination pattern when the many muscle of a limb allow for multiple solutions?

I revisit this problem from the emerging perspective of neuromechanics that emphasizes finding and implementing families of feasible solutions, instead of a single and unique optimal solution. Those families of feasible solutions emerge naturally from the interactions among the feasible neural commands, anatomy of the limb, and constraints of the task. Such alternative perspective to the neural control of function is not only biologically plausible, but sheds light on the most central tenets and debates in the fields of neural control, robotics, rehabilitation, and brain-body co-evolutionary adaptations. This perspective developed from courses I taught to engineers and life scientists at Cornell University and the University of Southern California, and is made possible by combining fundamental concepts from mechanics, anatomy, robotics and neuroscience with advances in the field of computational geometry.

Fundamentals of Neuromechanics is intended for neuroscientists, roboticists, engineers, physicians, evolutionary biologists, athletes, and physical and occupational therapists seeking to advance their understanding of neuromechanics. Therefore, the tone is decidedly pedagogical, engaging, integrative and practical to make it accessible to people coming from a broad spectrum of disciplines. I attempt to tread the line between making the mathematical exposition accessible to life scientists, and convey the wonder and complexity of neuroscience to engineers and computational scientists. While no one approach can hope to definitively resolve the important questions in these related fields, I hope to provide you with the fundamental background and tools to allow you to contribute to the emerging field of neuromechanics.

Engineering

Theoretical framework

http://valerolab.org/fundamentalsAll lectures are available online

Free e-Book at universities with SpringerLink

http://valerolab.org/fundamentals/

http://valerolab.org/fundamentals

How did this paradox arise?

The brain is confronted with controlling muscles and tendons system, yet for mathematical and technical

convenience (both very good reasons!) we have historically phrased the problem as one of torque

control or of simplified musculature.

Much of what I have learned has come from using and extending techniques to study the mechanics of

tendon-driven systems.

What is a limb?What is limb function?

BrainSpinal cord

Muscles, bones and joints

Motor Cotex

Motorneuron

Extensor

Flexor

The musculo-skeletal system “filters” the propagation of neural commands

MusclesTendons

&joints

BonesMuscle

activation(activation space)

Muscleforce

(muscle force space)

Jointtorques

(torque space)

Endpointforces and torques

(wrench space)

Block diagram of transformations in a forward limb model

Input OutputMusclesTeTeT ndons

&joints

BonesMusclefofof rcrcr e

Joinini ttorqrqr ues

EnEnE dpdpd oinini tfofof rcrcr es and torqrqr ues

Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input OutputTeTeT ndons&

jointsBones



Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input OutputBonesEnEnE dpdpd oinini t

fofof rcrcr es and torqrqr ues

Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input Output

Motor Cotex

Motorneuron

Extensor

Flexor

100 7 Feasible Neural Commands and Feasible Mechanical Outputs

feasible activation set(nondimensional units)

feasible muscle force set(N)

feasible torque set(N-m)

F matrix0F matrix

R m

atrix

endpoint feasible wrench set(forces in N)

350

525

0

700

f1

f3

f2

1

1

0

1

a1

a3

a2

0 50 100

-75

-50

-25

0

25

τ2

τ3 τ1

Torque at joint 1

Torque at joint 2

J matrix-T

Force in x

Force in y

-50 -250 25

0

100

200

w1

w3

w2

(a) (b)

(c)(d)

Fig. 7.3 Sequence of linear transformations of the feasible activation set in a forward model ofstatic force production by a tendon-driven limb. Using the concepts in Figs. 7.1 and 7.2, we use asneural input a the feasible activation set (a positive unit cube) to produce b the feasible muscle, cfeasible joint torque, and d endpoint feasible wrench sets. Adapted with permission from [2]

point wrenches, respectively. Their positive linear combinations specify the set of allpossible outputs in those spaces. If we consider them as the vi vectors of Fig. 7.2,we can build the zonotopes that correspond to the feasible joint torque set and thefeasible output wrench set as shown in Fig. 7.3. In the specific simple case shownin Fig. 7.4, the endpoint wrench is a planar force vector—consisting of fx and fycomponents only. This zonotope is called the feasible force set, and not the feasiblewrench set because the endpoint output contains no torques (see Sect. 2.6).

Valero-Cuevas. Fundamentals of Neuromechanics, 2015

This propagation of neural

commands defines

feasible inputs and

outputs

Feasible actions are defined bythe anatomical routing of tendons

Spoor, An, Yoshikawa, Brand, Leijnse, Valero-Cuevas, etc.

350

700

525

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25

muscle force space(N)

joint torque space(N-m)

Tendon pathsand moment arms

Force in x

Force in y

xy

0f1

Torque at joint 1

f 3

f 2

τ2

τ3 τ1

Torque at joint 2

{(f1 * r1,1), (f1*r2,1))

{(f2 * r1,2), (f2*r2,2))

{(f3 * r1,3), 0)

R matrixR matrix

-50 -25 0 25

0

100

200

0

100

200

endpointwrench space(forces in N)

J matrixJ matrix-T

w w

w

1 3

2

m3

m1m2

r1,1r1,1

r1,3r1,3

r1,2r1,2 r2,2r2,2

r2,1r2,1

ground

joint 1

joint 2



350

700

525

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25




Force in x

Force in y

xy

0f1

Torque at joint 1

f 3

f 2

τ2

τ3 τ1

Torque at joint 2

{(f1 * r1,1), (f1*r2,1))

{(f2 * r1,2), (f2*r2,2))

{(f3 * r1,3), 0)

R matrixR matrix

-50 -25 0 25

0

100

200

0

100

200


J matrixJ matrix-T

w w

w

1 3

2

m3

m1m2

r1,1r1,1

r1,3r1,3

r1,2r1,2 r2,2r2,2

r2,1r2,1

ground

joint 1

joint 2



350

700

525

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25




Force in x

Force in y

xy

0f1

Torque at joint 1

f 3

f 2

τ2

τ3 τ1

Torque at joint 2

{(f1 * r1,1), (f1*r2,1))

{(f2 * r1,2), (f2*r2,2))

{(f3 * r1,3), 0)

R matrixR matrix

-50 -25 0 25

0

100

200

0

100

200


J matrixJ matrix-T

w w

w

1 3

2

m3

m1m2

r1,1r1,1

r1,3r1,3

r1,2r1,2 r2,2r2,2

r2,1r2,1

ground

joint 1

joint 2



350

700

525

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25




Force in x

Force in y

xy

0f1

Torque at joint 1

f 3

f 2

τ2

τ3 τ1

Torque at joint 2

{(f1 * r1,1), (f1*r2,1))

{(f2 * r1,2), (f2*r2,2))

{(f3 * r1,3), 0)

R matrixR matrix

-50 -25 0 25

0

100

200

0

100

200


J matrixJ matrix-T

w w

w

1 3

2

m3

m1m2

r1,1r1,1

r1,3r1,3

r1,2r1,2 r2,2r2,2

r2,1r2,1

ground

joint 1

joint 2

Building a feasible torque set for a “complex” limb

Valero-Cuevas, 2005. J Biomech.

r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3

+fingertip flexors produce

positive torque

DOF 1

DOF 2



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

Quad. II Quad. I

Quad. III Quad. IV

m3

r1,4r2,4

flexion torqueextension torque

flexion torque

extension torque



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

m3



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

m3



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

m3



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

m3



r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ2

τ1

m5

m1

m4

m2

m3

One analytical starting point:A working definition of “versatility”

Simply put: the ability to produce end-point force in every direction—i.e., controllability.

Not to worry, it can be extended to motion in every direction!

Lucía Valero

Valero-Cuevas. A mathematical approach to the mechanical capabilities of limbs and fingers. 2009.

Versatility ≡ feasible torque and force sets that include the origin

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25

feasible torque set(N-m)

endpointfeasible force set

(forces in N)

τ 2

τ 3 τ 1

Torque at joint 1

Torque at joint 2

Force in x

Force in y

-50 -25 0 25

0

100

200

0

100

200

ww

w

1

3

2

That is, producing end-point force in every Cartesian direction requires that you produce torques in every direction in “torque space”

Valero-Cuevas 1998, 2005, 2009

convex sets remain convex under linear mapping

All’s well

m3

m1m2

ground

All’s well

m3

m1m2

ground

...but

How many muscles do you need to include the origin in torque/force space?

-100 -50 0 50 100

-100

-50

50

-

-50

0

50

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25

feasible torque sets(Newton-meters)

τ 2

τ 3 τ 1

Torque at joint 1

Torque at joint 2

Torque at joint 1

Torque at joint 2

m3

m1m2

r1,1

r1,3

r1,2 r2,2

2-link limb with 3 muscles (N+1)2-link limb with 4 muscles (2*N)

r2,1

ground

m3

m1m2

r1,1

r1,3

r1,2 r2,4

r2,1

ground

m4r1,4

At least N+1 well-routed muscles

Wait a minute... but N+1 > N


Wait a minute... but N+1 > N...so you need more muscles than degrees of freedom?


Wait a minute... but N+1 > N...so you need more muscles than degrees of freedom?A versatile feasible torque set implies muscle redundancy for

submaximal outputs!

0 50 100

-75

-50

-25

25

-75

-50

-25

0

25

τ 2

τ 2

τ 3

τ 1

τ 1

Torque at joint 1

Torque at joint 2


Muscle redundancy is not an accident of evolution, but rather an appropriate structural

adaptation for versatility.

(and later we will see how having more muscles allows us to meet more function constraints)

Thus, versatile tendon-driven systems require “over-actuation”

And each tendon contributes in unique waysτ

2

τ1

FTS 1,2,3,4,5

FTS 1,2,3,4

FTS 1,2,3

FTS 1,2r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ1

FTS 1,2,3


2

τ1

FTS 1,2,3,4,5

FTS 1,2,3,4

FTS 1,2,3

FTS 1,2

Tendons define the size and shape of the feasible torque and feasible force sets

r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ1

FTS 1,2,3


2

τ1

FTS 1,2,3,4,5

FTS 1,2,3,4

FTS 1,2,3

FTS 1,2

Tendons define the size and shape of the feasible torque and feasible force sets

So which muscle would you give up?

r1,2

r1,5 r1,3r2,5

r2,4

5 muscles

r2,1r1,4

r1,1 m1

m4

m2

m5m3


positive torque

DOF 1

DOF 2

τ1

FTS 1,2,3

Redundancy does not imply robustness

With Jason KutchJ Biomech 2011

Now assistant professor at USC

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011

MUSCLE 1

MUSCLE 2

MUSCLE 3

Dimensionality and structure of the feasible input (solution space) for a given task

Computational GeometryVertex Enumeration (dual of Linear programming)

No cost function needed, simply description of feasible inputs and outputs!

feasible output spacefeasible input space

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2





MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2





All possible ways to do

a same task

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011

MUSCLE 1

MUSCLE 2

MUSCLE 3

Some muscles are more redundant than others!

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2

Allo

wab

le Te

nsio

n Ra

nge

Maximal

0 MUSCLE1 2 3


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2

Allo

wab

le Te

nsio

n Ra

nge

Maximal

0 MUSCLE1 2 3

!"#$%&'()'#$%*%'

+,*-.%*'"*'/(#'

/%-%**0&1


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2

Allo

wab

le Te

nsio

n Ra

nge

Maximal

0 MUSCLE1 2 3

2$"*'+,*-.%'"*'/%-%**0&133

Allo

wab

le Te

nsio

n Ra

nge

!"#$%&'()'#$%*%'

+,*-.%*'"*'/(#'

/%-%**0&1


Can we show this for real systems?

Joining the XVI and XXI centuries

Valero-Cuevas et al, 2000-presentAnatomy Lesson of Dr.TulipRembrandt 1632

.Turn to age-old physical testing because biomechanical modeling remains a challenge

Robotic Arm

Extension Spring

Safety Shield

Fixation Device

Dynamometer

Figu

Mechanical Engineering

Mathematics

Computer ScienceHand SurgeryNeuroscience

Record the real input-output mapping

MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input Output

Robotic Arm

Extension Spring

Safety Shield

Fixation Device

Dynamometer

Figure 2

A. Constrain only radial force

LUM FPI FDP FDI FDS EDC EIP0

100

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)

B. Constrain only radial and dorsal force


1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)


1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)

C. Constrain radial, dorsal, and distal force (produce a well directed force)

What is the set of all muscle actions to produce a given force?

A subset of 7-dimensional space!

Kutch & Valero-Cuevas, 2011



100

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)



1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)


1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)







100

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)



1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)


1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)







100

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)



1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)


1

Muscle

Allo

wab

le T

ensi

on R

ange

(% M

ax)


Necessary

Necessary & Intolerant to dysfunction





What is the nature and structure of feasible sets?

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension


MUSCLE 1

MUSCLE 2

MUSCLE 3

A first approach: the bounding box

MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension


MUSCLE 1

MUSCLE 2

MUSCLE 3


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2


MUSCLE 1Tension

MUSCLE 2Tension

MUSCLE 3Tension

Max tension

Max tension

Max tension

Max tension

Max tension

Constraint 1

Constraint 2


MUSCLE 1

MUSCLE 2

MUSCLE 3

Constraint 1

Constraint 2

Allo

wab

le Te

nsio

n Ra

nge

Maximal

0 MUSCLE1 2 3


Chapter 9


Use computational geometry to find the structure of these high-dimensional subspaces8.3 Vertex Enumeration in Practice 121

unit N-cube convex polytopevertices of

convex polytope

Feasibleinputset

Feasibleoutput

set

(a) (b) (c)

linearmapping

Fig. 8.5 Top row Schematic representation of Algorithm 1. Note that the addition of inequalityconstraints reduces the size and distorts the shape of the original feasible input set—the positive unitN-cube—and transforms it into a new convex polytope. The more faces the convex polytope hasthe more vertices are required to define it. Columns a–c: Schematic representation of Algorithm 2.As illustrated in Fig. 7.6, the projection of the feasible input set produces a feasible output set thatis a convex hull that defines the feasible muscle force, joint torque, wrench, or output force sets,depending on the framing of the problem. Figure8.1 shows a real-world example of mapping afeasible activation set in R7 onto the feasible force set in the plane of finger flexion in R2. Note thatthe vertices that become internal points of the feasible output set hint at the shape of the feasibleinput set, which is hard to describe or visualize given its high-dimensionality. This is an area ofactive research [18]

upon the most necessary points, and the reader is encouraged to see the referencesmentioned.

Recall that in Sect. 7.6 we introduced two types of representations of a convexpolytope. In this case, we are interested in finding the vertices of a high-dimensionalconvex polytope defined by a set of linear inequality constraints [15] as in

Ax ≤ b (8.6)

where A ∈ RM×N is a matrix containing M equations in N variables, x ∈ RN , andb ∈ RM is a vector of constants.

Do you notice the similarity to the phrasing of the canonical form of the linearprogramming problem in Eq.5.9? Notably, however, there is no need to have a costfunction. That is, we are not interested in finding the maximal force in a particulardirection in the plane of finger flexion as in Eq.5.22. Instead, we are interested infinding all possible maximal forces in the plane of finger flexion.

8.3 Vertex Enumeration in Practice 121

unit N-cube convex polytopevertices of

convex polytope

Feasibleinputset

Feasibleoutput

set

(a) (b) (c)

linearmapping

Fig. 8.5 Top row Schematic representation of Algorithm 1. Note that the addition of inequalityconstraints reduces the size and distorts the shape of the original feasible input set—the positive unitN-cube—and transforms it into a new convex polytope. The more faces the convex polytope hasthe more vertices are required to define it. Columns a–c: Schematic representation of Algorithm 2.As illustrated in Fig. 7.6, the projection of the feasible input set produces a feasible output set thatis a convex hull that defines the feasible muscle force, joint torque, wrench, or output force sets,depending on the framing of the problem. Figure8.1 shows a real-world example of mapping afeasible activation set in R7 onto the feasible force set in the plane of finger flexion in R2. Note thatthe vertices that become internal points of the feasible output set hint at the shape of the feasibleinput set, which is hard to describe or visualize given its high-dimensionality. This is an area ofactive research [18]

upon the most necessary points, and the reader is encouraged to see the referencesmentioned.

Recall that in Sect. 7.6 we introduced two types of representations of a convexpolytope. In this case, we are interested in finding the vertices of a high-dimensionalconvex polytope defined by a set of linear inequality constraints [15] as in

Ax ≤ b (8.6)

where A ∈ RM×N is a matrix containing M equations in N variables, x ∈ RN , andb ∈ RM is a vector of constants.

Do you notice the similarity to the phrasing of the canonical form of the linearprogramming problem in Eq.5.9? Notably, however, there is no need to have a costfunction. That is, we are not interested in finding the maximal force in a particulardirection in the plane of finger flexion as in Eq.5.22. Instead, we are interested infinding all possible maximal forces in the plane of finger flexion.

Brian Cohn May Szedlák

with Fukuda K and Gärtner B

Valero-Cuevas et al., 1998 - 2015

The Hit-and-Run algorithm can sample such subspaces in up to 40 dimensions!

We can now sample subspaces in up to 40

dimensions to provide a statistical description of

their structure

9.5 Probabilistic Neural Control 151

a1 a3

a2

p

a1 a3

a2

p

q

a1

a1

a3

a3

a2

a2

a1 a3a2

pp1

a1 a3

a2

0

0

0

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p1

(a) (b)

(c)

(e)

(f)

(d)

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lity

dens

ityP

roba

bilit

y de

nsity

Activation Activation Activation

1 N Output

2 N Output

0

K r

s

0

0

0 018.06.02.0 4.0 018.06.02.0 4.0 6.02.0 4.0 0.8 1

0 018.06.02.0 4.0 018.06.02.0 4.0 18.06.02.0 4.0

3.04.04.0

2.0

40

20

40

10.5

25

20

2.0 1.5

1

1

11

1

1

1

1

11

1

1

Fig. 9.7 The Hit-and-Run algorithm applied to a convex body K defined as the 2D shaded polygonembedded in 3D. aYou begin by finding a valid point p0 ∈ K , and run a linear program in a randomvector direction q, (b) to find the extreme points r and s at both ends. c You then use a uniformprobability distribution to sample a point p1 along the line r s, and d use the point p1 to repeat step(a) iteratively to find the set of points pi for i = 0, . . . ,M , where M is large enough to guaranteeconvergence to a uniform sampling of K . For an example of such iterative stochastic sampling seeFig. 9.8. e The histograms of the sampled points pi are an approximation to the probability densityfunctions of the convex body K for a 1 N target output force magnitude by the limb in a particulardirection. f If we change the target magnitude of force output to 2 N, the probability distributionschange where now a2 cannot be fully activated, a1 moves toward full activation, and the plane’ssurface area is centrally distributed about a3

Submaximal force

Near maximal force

Applied to a 7-muscle finger model producing static force

Valero-Cuevas et al 1998

45'*(.,6(/'*70-%'

%+8%99%9'

"/':5

Histograms (Bayesian priors) of activation to produce static force of different magnitudes

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with associated cost for each solution

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Summary about muscle redundancy

• Redundancy does not imply robustness


• Redundancy does not imply robustness• Every muscle contributes uniquely to function


• Redundancy does not imply robustness• Every muscle contributes uniquely to function•More muscles allow more complex tasks


• Redundancy does not imply robustness• Every muscle contributes uniquely to function•More muscles allow more complex tasks• We have barely enough muscles for real-world tasks!


The nature and structure of feasible sets

•Co-contraction is often not an option, and in fact loses meaning



•Agonist-antagonist language loses meaning



•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant



•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!




•Critical evaluation of synergies




•Critical evaluation of synergies•The statistical structure of the solution space shows the path to probabilistic motor control





•Motor control is a Big Data problem












Conclusionsaboutdesignof tendon-drivensystems

• Thehumanhandhascri$calmorphologicalfeatureslendingitverygoodgraspcapabili6es.

• Thegraspingcapabili$esofrobo6chandscanbedras$callyimprovedbyexploringthefulldesignspace

–Non-uniformmaximaltendontension

distribu6ons

–Centerofrota6onsnotinthemiddleofthejoint

52

Roadmap to computational exercises

Brian Cohn

1

ISBN 978-1-4471-6746-4


Francisco J. Valero-Cuevas

Fundamentals of NeuromechanicsFundamentals of Neuromechanics

Valero-Cuevas


Francisco J. Valero-CuevasFundamentals of Neuromechanics

This book provides a conceptual and computational framework to study how the nervous system exploits the anatomical properties of limbs to produce mechanical function. The study of the neural control of limbs has historically emphasized the use of optimization to find solutions to the muscle redundancy problem. That is, how does the nervous system select a specific muscle coordination pattern when the many muscle of a limb allow for multiple solutions?

I revisit this problem from the emerging perspective of neuromechanics that emphasizes finding and implementing families of feasible solutions, instead of a single and unique optimal solution. Those families of feasible solutions emerge naturally from the interactions among the feasible neural commands, anatomy of the limb, and constraints of the task. Such alternative perspective to the neural control of function is not only biologically plausible, but sheds light on the most central tenets and debates in the fields of neural control, robotics, rehabilitation, and brain-body co-evolutionary adaptations. This perspective developed from courses I taught to engineers and life scientists at Cornell University and the University of Southern California, and is made possible by combining fundamental concepts from mechanics, anatomy, robotics and neuroscience with advances in the field of computational geometry.

Fundamentals of Neuromechanics is intended for neuroscientists, roboticists, engineers, physicians, evolutionary biologists, athletes, and physical and occupational therapists seeking to advance their understanding of neuromechanics. Therefore, the tone is decidedly pedagogical, engaging, integrative and practical to make it accessible to people coming from a broad spectrum of disciplines. I attempt to tread the line between making the mathematical exposition accessible to life scientists, and convey the wonder and complexity of neuroscience to engineers and computational scientists. While no one approach can hope to definitively resolve the important questions in these related fields, I hope to provide you with the fundamental background and tools to allow you to contribute to the emerging field of neuromechanics.

Engineering

Theoretical framework


Chapters 7, 8 and 9



Define your input-output relation

MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input OutputMusclesTeTeT ndons

&joints

BonesMusclefofof rcrcr e



Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input OutputTeTeT ndons&

jointsBones



Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input OutputBonesEnEnE dpdpd oinini t

fofof rcrcr es and torqrqr ues

Outptpt ut

Motor Cotex

Motorneuron

Extensor

Flexor


MusclesTendons

&joints

BonesMuscle


Muscleforce


Jointtorques

(torque space)


(wrench space)


Input Output

Motor Cotex

Motorneuron

Extensor

Flexor

Find the bounding box of the feasible static force of a 31-muscle 3D cat hindlimb

144 9 The Nature and Structure of Feasible Sets

X

Y TPPTPLPB

FDLFHLLG

SOL*MG

*PLAN

TAEDL*

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BFASM

PSOASPECADFADL

GRACSART

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MGBFP

VLGRACVIRFLGEDLSMPLANVMST

FHLMG

SOLLG

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FDL

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−2.00 0.00 2.00

Moment arm (cm) Flex/Ext

−2.00 0.00 2.00

Moment arm (cm) Add/Adb

Z

Y

RFPECADLPYR

GRACBFAADFBFPQF

SARTPSOASSMSTGMAXGMEDGMIN

−2.00 0.00 2.00

Hip Int/Ext Rot (cm)

a. Sagitatal view of right hind limb b. Frontal view of right hind limb

Knee

Ankle

Hip

Fig. 9.3 Moment arms of the 7 DOF, 31 muscle cat hindlimb model [2]. Figure adapted withpermission from [70]

of a cat (Felis catus) with 31 muscles actuating 7 kinematic DOFs from the hip tothe ankle,6 Fig. 9.3.

One application of the techniques described in this book was to find feasible forcesets7 for models of the hindlimb of the cat [71]. A subsequent study explored thebounding box of the feasible activation set for force in one 3D direction [2]. Theyfound that, as the magnitude of the force in that direction increased, the feasibleranges for many muscles remain large even when approaching maximal feasibleforce.

We studied the bounding box of the feasible activation sets for force production inevery 3D direction, also at different force magnitudes [70] because, as motivated bythe previous section, understanding neural control strategies requires that we knowas much as possible about the global properties and structure of feasible activationsets.

In real life, the neural control of musculature must regulate the magnitude and thedirection of force vectors, as in locomotion [72] and manipulation [73]. Therefore,it is of interest how the structure of the feasible activation set changes as you varythe direction of force production.

In response to this need, we developed the vectormap approach as shown inFig. 9.4 [70]. This technique allows one to visualize the properties of the enclosedfeasible set more intuitively. More importantly, by mapping the properties of theenclosed feasible set onto the surface of the sphere, several calculations can be done

6The general model has 7 DOFs, but in this analysis hip ad- ab-duction was frozen so the model isreally a 6 DOF model.7That is, constraints were added to enforce that the torque elements of the output wrench be zero.

X

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TAEDL*

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*RFPSOAS

SART

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VM*RF

SM*LGPLAN*MG*GRAC*BFPST*

BFASM

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GRACSART

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MGBFP

VLGRACVIRFLGEDLSMPLANVMST

FHLMG

SOLLG

PLANTP

FDL

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ï��

Moment arm (cm) Add/Adb

Z

Y

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ï��

Hip Int/Ext Rot (cm)

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Knee

Ankle

Hip

Implement Vectormap

9.4 Vectormap Description of Feasible Sets 145

10 20 30 40 50 60Feasible force output for a given direction (N)

2D Feasible force set 3D Feasible force set Spherical vectormap

60N{

dorsal

ventral

ante

rior

post

erio

r

(a) (b) (c)

Fig. 9.4 Vectormap visualization for a cat hindlimb in a static force production task. a Take theexample of a 2D feasible force set, where that polygon is enclosed in a circle. The thin black linesemanating from the origin are the lines of action of each of the 31 muscles. The distance from theorigin to the boundary of the feasible force set (i.e., the maximal force in that direction) is assigneda color, blue for small and yellow for large, as shown for the eight rays emanating from the origin.That color-code is vectormapped onto that point on the circle as shown for the posterior, anterior,dorsal and ventral directions, and 4 others in between. b The same can be done for a 3D feasibleforce set that is a polyhedron enclosed in a sphere (only the cross-section of the sphere is shown).c The color on the surface of the sphere now contains all the information of the enclosed feasibleforce set—but in a more intuitive way that can be compared across directions and 3D feasible sets.Adapted with permission from [70]

on that manifold. For example, one can find the difference, sum, or average of twosuch feasible sets. For example, Fig. 9.5 shows the average and standard deviationmaximal feasible force from three different cat models with different limb anatomyand moment arms. The vectormap approach is novel and useful because it allows usto generate cross-species, inter-species, and inter-task comparisons.

This same methodology can be used to represent feasible activations. Take forexample the feasible activations associated with the spherical vectormap of the fea-sible force set shown in Fig. 9.4c. For each 3D direction of the feasible force set,that muscle is associated with a particular activation level. These values are foundfrom the optimal, unique coordination patterns that produce maximal force in eachof those directions mapped onto the spherical vectormap of the feasible force set. Ifyou collect all such unique values, they are in fact a 3D object, whose vectormapcan be shown in 3D. Consider the case of one single muscle, say the vastus lateralis,as shown in the top row of Fig.9.6. The large vectormap at the far right shows the

Species average

5

10

15

20

25

30

35

40

45

50

55

dorsalposterior

Force (N)

Mean

SD

Same orientation as above

medial anteriorventral

lateral

Orientation

Implement Vectormap9.4 Vectormap Description of Feasible Sets 147

Vastus LateralisLower bounds

Muscle activation0 1.00.5

Upper bounds

α = 50% α = 60% α = 70% α = 80% α = 90% α = 100% (fmax)

Medial Gastrocnemius

Soleus

dorsal

posterior lateral

medialventral

anterior

Fig. 9.6 Vectormaps of the feasible activations sets for sub-maximal and maximal force out-put in every 3D direction. The vectormaps for 5 different sub-maximal force levels (α =50, 60, 70, 80, and 90%) for each of 3 muscles are shown, plus the larger vectormap for the maxi-mal feasible force magnitude. The top and bottom rows are, for each muscle respectively, obtainedfrom the maximal and minimal values of their bounding boxes. Figure adapted with permissionfrom [70]

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!NFG'SB!@

The Whitaker FoundationBiomedical Engineering Research Grant

The National Institutes of HealthNIAMS/NICHD R01-AR050520; R01-AR052345: R21-HD048566

Alexander von Humboldt FoundationMax Plank Institute for Human Brain and Cognitive

Sciences

Wenner-Gren FoundationKarolinska Institute

Department of EducationNIDRR OPTT-RERC

Documents

2016 CoSMo - CompNeuroscicompneurosci.com/wiki/images/8/83/Francisco_2016_CoSMo.pdf · Frame computational neuromechanics as a Big Data problem. Hot topics of the day Muscle redundancy